5 S 2 tot = S 2 T (y, t) + S 2 (y) = const. Z 2 (4.22) σ 2 /4 y = y z y t = T/T 1 2 (3.9) (3.15) s 2 = A(y, t) B(y) (5.1) A(y, t) = x d 1+α dx ln u 1 ] 2u ψ(u), u = x(y + x 2 )/t s 2 T A 3T d S 2 tot S 2 Z ()] d α 1 B(y) y 3 B(y) = c z y y = s 2 t = t c (y = ) s 2 = T A S 2 3T d (, t T c) = A(, t c ) (5.2) s 2 s 2 < y s 2 y y = S 2 y > tot (5.1) y t s 2 (5.1) 2 y B(y) 78
d > 2 α B(y) y (5.1) s 2 = c z = x d 1+α dx ln u 1 ] 2u ψ(u) c z y 1 d 2 + α c z 5.1 3 y t y y t- s 2 2 s 2 > d > 2 T c y T c y = T t c = T c /T 1 (3.12) (5.2) t c s 2 s 2 = 1 2 + α C νt ν c = T A S 2 3T d (, t i T c), ν = (d + α)/(2 + α) (5.3) 4 t c σs 2 (4.17) σ 2 s 4 = 5T d (2 + α)t A C ν t ν c (5.4) d 2 y t y y t y t 4 ln(t2/3 /y) y, 2 t s 2 4 ln(t2/3 /y) y ln(1/y) + 1], 2 2 = πt 2y y ln(1/y) + 1], 1/2 2 1 πt 2y 1/2 πy1/2, 1 79
s 2 < s 2 y = y s 2 < y y y > y(t) y = y() y s 2 y d 2 + α, d > 2 α s 2 y = 2 ln(1/y ) + 1], 2, 1 πy 1/2, 1 3 2 y 5.2 3 3 d = 3, α = 1 (5.1) y s 2 = u = x(y + x 2 )/t x 3 dx ln u 1 ] 2u ψ(u) c z y (5.5) α = s 2 s 2 1 s 2 2 y 5.2.1 3 s 2 > t c t = t c y = (5.5) A(, t c ) = dxx 3 ln u c 1/2u c ψ(u c )] = s 2, u c = x 3 /t c (5.6) (4.17) s 2 = A(, t c ) = c z y 1 σs 2 (5.6) t c = T c /T 8
σ s t c 1 (3.12) 1 3 C 4/3t 4/3 c s 2 = c z y 1 σ 2 s, y 1 = T A 6c z T σs 2 T c σs 2 = 2C ( ) 4/3 4/3T Tc (5.7) T A T (5.5) (5.6) y A(y, t) c z y = A(, t c ) (5.8) y y 2 t t c t = T/T = t c T/T c y T/T c t c 1 y(t) = y 1 (t)σ 2 (t) (5.9) y = y 1 (t)σ 2 + σ 2 (t)] y (3.13) 4 x 3 dx ln u 1 ] 2u ψ(u) 1 3 C 4/3t 4/3 πt c y (5.1) 4 (5.1) (5.8) y y y t πt c C 4/3 y 4 3 (t4/3 t 4/3 c ) y (T/T c 1) 2 t c { 4C4/3 t 1/3 ( ) ]} 4/3 2 c T y(t) 1 3π T c = { 16C4/3 t 1/3 ( ) } 2 c T 1 9π T c ( ) 2 ( ) 2 16C4/3 T t 2/3 c 1 (5.11) 9π T c (4.27) y 1 (t) σ 2 (t) σ 2 (t) = y (t) y 1 (t) C 4/3t 4/3 ( c T 6c z y 1 T c ) 4/3 1] 2C 4/3t 4/3 c 9c z y 1 (T/T c 1) ( ) T 1 T c (5.12) 81
5.2.2 s 2 y y (5.5) 1 y = s 2 /c z t c = T c /T t c A(, t c) = s 2 = c z y (5.13) A(, t c ) = c z y 1 σ 2 s σ p y = y() = y 1 σ 2 p, y 1 = T A 6c z T (5.14) (4.18) : y(σ, ) = y 1 (σ 2 σ 2 s), : y(σ, ) = y 1 (σ 2 + σ 2 p), t c 1 A(, t c) t c 4/3 Tc = t ct σ p σp 2 = 2C ( ) 4/3T T 4/3 c (5.15) T A T (5.13) (5.5) y(t) y T/T c A(y, t) c z y = A(, t c) (5.16) t c t u ψ(u) ln u 1 2u ψ(u) 1, (1 u) 12u2 (5.5) t 2 x 3 dx ln u 1 ] 2u ψ(u) = dx x3 12u 2 = t2 12 x dx (y + x 2 ) 2 t 2 24y(1 + y), u = x(y + x2 )/t y t y(t) y + t2 = y + (t c) 2/3 ( ) 2 T 24c z y 8C 4/3 y t 2-1/y T c 82
Curie-Weiss t (5.8) (5.16) y t y t y 1.4 y(t).2 T c /T =.1.5.1. 1. 2. 3. 4. 5. T/T c 1: y 11 (5.8) (5.16) Appendix A.4 5.2.3 3 χ(q) y (5.1) d = 3, α = s 2 = A(y, t) c z y (5.17) A(y, t) = x 2 dx ln u 1 ] 2u ψ(u), u = (y + x 2 )/t u x x x 3 x 2 s 2 y T N y = s 2 83
.6 T c * /T =.1.4 y(t).5.2.1.. 1. 2. 3. 4. 5. * T/T c 11: y A(, t N ) = dxx 2 ln u c 1/2u c ψ(u c )] = s 2, u c = x 2 /t N (5.18) T t N = T N /T t N 1 (3.13) 1 2 C 3/2t 3/2 N s2 (5.19) (5.17) u x (4.17) σ 2 s 4 = 15T 2T A C 3/2 t 3/2 N (5.2) Curie-Weiss T (5.17) y t y t = T/T Q Curie-Weiss (5.17) Appendix A.4 84
(t t N ) y y y x 2 dx ln u 1 ] 2u ψ(u) 1 2 C 3/2t 3/2 πt N y (5.21) 4 (5.17) y 1 y y y t πt N C 3/2 y 4 2 (t3/2 t 3/2 N ) ( ) 2 ( ) ] 3/2 2 2C3/2 T y t N 1 π T N ( 3C3/2 π ) 2 ( ) 2 T t N 1 (5.22) T N χ(q) (T/T N 1) 2 5.2.4 s 2 < s 2 1 T N = t N T σ p A(, t N ) = s 2 = c z y, σ 2 p = y y 1 (5.23) y = y() y 1 4 (5.23) (5.17) A(y, t) c z y = A(, t N) (5.24) y t (5.24) (5.23) y t 2 (5.17) u x 2 dx ln u 1 ] 2u ψ(u) dx x2 12u 2 = t2 x 2 dx 12 (y + x 2 ) 2 ( = t2 ) dx x2 12 y y + x 2 = t2 1 y tan 1 1 1 ] πt2 24 y 1 + y 48 y 85
(5.24) y ( ) 2 y y + πt2 2 T = y + 48c z y 48 t N 5/4 c z C 3/2 T N 5.3 2 2 3 (t > ) y log y 1 t = y > y 5.3.1 y t s 2 = x 2 dx ln u 1 ] 2u ψ(u) c z y (5.25) u = x(y + x 2 )/t c z = 1 (t > ) y > s 2 y (5.25) s 2 > t = y y t 2/3 y (5.25) s 2 = t 4 ln ( t 2/3 y ) c z y y t c = T c /T t c = 2 s 2 y t 3/2 exp( 2t c /t) y t c 3 2 3 2 3 (Takahashi 1997) 86
s 2 < y y y = s 2 /c z 3 = t2 24 x 2 dx ln u 1 ] 2u ψ(u) 1 y y tan 1 1 y + 1 1 + y x 2 dx 1 12u 2 ] πt2 48y, (y 1) y y 1 y c z y = c z y + πt2 48y y y y y 3 T 2 πt 2 y = y + 48c z y y 5.3.2 y y (5.25) s 2 = t 2 φ(y/t) φ((1 + y)/t)] 1 2 y ln φ(u) = xdx ln u 1 ] 2u ψ(u) ( ) ] 1 + 1 y = (u 1/2) ln u + u + ln Γ(u) ln 2π, u = (y + x 2 )/t (5.26) (5.26) φ(u) u u 1 ln u, u 1 φ(u) 2 (5.27) 1 12u, u 1 s 2 < y y u = (y + x 2 )/t 1 5.26 y s 2 y 2 ( ) ] 1 ln + 1 = s 2 y 87
(5.26) ( ) ] ( ) ] 1 1 y ln + 1 y ln + 1 = t φ(y/t) φ((1 + y)/t)] (5.28) y y (5.27) u 1 φ(u) u y y ( ) ] ( 1 1 y ln + 1 y ln y y = y + y ) ] ( ) 1 + 1 ln (y y ) t2 y 12y t 2 12y ln(1/y ) + t 2 ln(1/y ) s 2 > y y/t 1 φ(u) u 1 u y s 2 = t 4 ln ( t y ) 12 ( ) ] 1 y ln + 1 y y y t N = T N /T ( ) ] 1 t N = 2 s 2 = y ln + 1 y y y te 2t N /t t < t N 5.4 3 (5.8) T c (5.16) (5.8) SCR c z SCR 2 (5.8) y c z 4 88
Arrott (5.8) y c z y (5.8) (5.8) y t (5.8) T T A T A T t 5.4.1 p eff /p s vs t c Curie-Weiss Curie p eff p s p eff /p s ( σ eff, σ s ) 1 T c p eff p c (p c + 2) = p 2 eff p c p s (Rhodes & Wohlfarth 1963) p c p s p c /p s T c Rhodes-Wohlfarth T c p c /p s = 1 T c p c /p s > 1 2 Takahashi Rhodes-Wohlfarth p c /p s p eff /p s T c t c = T c /T 89
(Takahashi 1986) χ Curie-Weiss (gµ B ) 2 χ = N µ 2 B σ2 eff 3(T T c ) y χ = N /(2T A y) Curie-Weiss y t χ N = = 1 2T A y = µ 2 B σ2 eff 3(gµ B ) 2 (T T c ) σ 2 eff 12T (t t c ) (5.7) σ 2 s t c σeff 2 = 12T (t t c ) 2T A y (σ eff /σ s ) 2 2 + α 1 1C ν (dy/dt) = 6 σ2 s 4 1 (2 + α) 15C ν t ν c ( ) t tc (5.16) y Curie-Weiss (t t c ) t- 3 t c 1 dy/dt.15 (α = 1, ν = 4/3) σ eff /σ s t c = T c /T t ν c σ eff /σ s 1.4 t 2/3 c (5.29) (5.29) 12 T ZrZn 2 (Kontani, et al. 1975), MnSi(Yasuoka, et al. 1978), Ni 3 Al(de Boer, et al. 1969), Sc 3 In(Hioki & Masuda 1977), Y(Co,Al) 2 (Yoshimura, et al. 1987), (Fe,Co)Si(Shimizu et al. 199), YNi x (Nakabayashi, et al. 1992) σ eff /σ s T c /T T NMR F 1 (5.16) (Nakabayashi et al. 1992) y σ s σ eff T c (K) T (K) σ eff /σ s T c /T MnSi.4 2.25 3 231 5.6.194 Ni 3 Al.77 1.3 41.5 276 16.9.15 Sc 3 In.45 1.3 5.5 479 28.9.115 ZrZn 2.12 1.44 21.3 139 12..15 9: 9
8 6 Y(CoAl) 2 (FeCo)Si YNi σ eff /σ s 4 2 Ni3Al Sc3In ZrZn2 MnSi.5.1.15 T c /T 12: σ eff /σ s - T c /T YNi 2 Rhodes-Wohlfarth Takahashi 13 x, y Rhodes-Wohlfarth Takahashi 8 6 2 p C /p s 4 p eff /p s 1 2 1 2 T c.5.1.15 T c /T 13: Y x Ni y p eff /p s T c /T (T c /T ) 2/3 (5.29) Y(Co,Al) 2 (p eff /p s ) 2 (T /T c ) 4/3 14 T F 1 p eff /p s T c /T 91
Y(Co Al) 2 4. (p eff /p s ) 2 2... 1. 2. (T /T c ) 4/3 14: Y(CoAl) 2 p eff /p s T c /T (5.8) c z Rhodes-Wohlfarth Takahashi Rhodes-Wohlfarth Takahashi t c 1 t c = T c /T Curie-Weiss t c T σ eff /σ s t c T c T t c 1 Curie-Weiss S 2 i = S(S + 1) = 3 N 2 q dω coth βω 2 Imχ(q, ω) (5.3) T c J kt c τ h/j τ 92
J kt/ω (5.3) coth(βω/2) 2T/ω S(S + 1) = 3kT N 2 q dω π Imχ(q, ω) ω = 3kT N 2 Reχ(q, ) q Reχ(q, ) N S(S + 1) 3kT 5.4.2 Q 3 Γ q = Γ q α (κ 2 + q 2 ) = 2πT x α (y + x 2 ) Γ q y Curie-Weiss 1 Γ q x α T = 2π y t y y/ t Γ q /x α T β-mn β-mn.9 Al.1 Q (Shiga, et al. 1994) 15 SCR (5.1) 93
5 4 β Mn β Mn.9 Al.1 Γ (K) 3 2 1 1 2 3 T (K) 15: β-mn Γ F. R. de Boer, et al. (1969). Exchange-Enhanced Paramagnetism and Weak Ferromagnetism in the Ni 3 Al and Ni 3 Ga Phases; Giant Moment Inducement in Fe-Doped Ni 3 Ga. J. Appl. Phys. 4:149 155. T. Hioki & Y. Masuda (1977). Nuclear Magnetic Resonance and Relaxation in Itinerant Electron Ferromagnet Sc 3 In. J. Phys. Soc. Japan 43:12 126. M. Kontani, et al. (1975). NMR Studies on Magnetic Properties of ZrZn 2. J. Phys. Soc. Japan 39(3):665 671. R. Nakabayashi, et al. (1992). Itinerant Electron Weak Ferromagnetism in Y 2 Ni 7 and YNi 3. J. Phys. Soc. Japan 61(3):774 777. P. Rhodes & E. P. Wohlfarth (1963). The Effective Curie-Weiss Constant of Ferromagnetic Metals and Alloys. Proc. R. Soc. 273:247 258. M. Shiga, et al. (1994). Polarized Neutron Scattering Study of β-mn and β-mn.9 Al.1. J. Phys. Soc. Jpn. 63(5):1656 166. K. Shimizu, et al. (199). Effect of Spin Fluctuations on Magnetic Properties and Thermal Expansion in Pseudobinary System Fe x Co 1 x Si. J. Phys. Soc. Jpn. 59(1):35 318. Y. Takahashi (1986). On the Origin of the Curie-Weiss Law of the Magnetic Susceptibility in Itinerant Electron Ferromagnetism. J. Phys. Soc. Jpn. 55:3553 3573. Y. Takahashi (1997). Spin-fluctuation theory of quasi-two-dimensional itinerant-electron ferromagnets. J. Phys.: Condens. Matter 9:1359 1372. H. Yasuoka, et al. (1978). NMR and Susceptibility Studies of MnSi above T c. J. Phys. Soc. Japan 44(3):842 849. 94
K. Yoshimura, et al. (1987). NMR Study of Weakly Itinerant Ferromagnetic Y(Co 1 x Al x ) 2. J. Phys. Soc. Jpn. 56:1138 1155. 95